A wider nonlinear extension of Banach-Stone theorem to 𝐶₀(𝐾,𝑋) spaces which is optimal for 𝑋=ℓ_{𝑝}, 2≤𝑝<∞

Abstract

It is proven that if X is a Banach space, K and S are locally compact Hausdorff spaces and there exists an (M, L)-quasi isometry T from C-0(K, X) onto C-0(S, X), then K and S are homeomorphic whenever 1 <= M-2 < S(X), where S(X) denotes the Schaffer constant of X, and L >= 0. As a consequence, we show that the first nonlinear extension of Banach-Stone theorem for C-0(K, X) spaces obtained by Jarosz in 1989 can be extended to infinite-dimensional spaces X, thus reinforcing a 1991 conjecture of Jarosz himself on epsilon-bi-Lipschitz surjective maps between Banach spaces. Our theorem is optimal when X is the classical space l(p), 2 <= p < infinity.